#8930007

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4546 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(2,5)(6,8)(9,10,15,12)(11,18,13,16)(14,20,22,17)(19,23,21,24)(25,26)(27,32)(28,31)(29,33)(30,36)(34,38)(35,37)(39,40);(3,7)(6,8)(9,11,15,13)(10,16,12,18)(14,19,22,21)(17,23,20,24)(25,27)(26,32)(28,29)(30,35)(31,33)(34,39)(36,37)(38,40);(1,2)(3,6)(4,5)(7,8)(10,12)(14,22)(16,18)(19,21)(26,28)(27,33)(30,38)(37,39);(9,12,15,10)(11,18,13,16)(14,20,22,17)(19,24,21,23)(25,28,31,26)(27,29,33,32)(30,36,38,34)(35,37,40,39);(1,3)(2,6)(4,7)(5,8)(9,13,15,11)(10,16,12,18)(14,21,22,19)(17,23,20,24)(25,29)(26,33)(27,28)(30,37)(31,32)(34,40)(35,36)(38,39);(1,2)(3,6)(4,5)(7,8)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8837203);